topic badge

2.05 Rationalising the denominator

Lesson

Rationalising the denominator is an expression that means to rewrite a fraction that has surds in the denominator in such a way that we only have an integer in the denominator. By convention this is the preferred way to write fractions involving surds; it makes adding fractions involving surds much more simple and in the time before calculators it made calculations by hand much easier. So how can we achieve this>?

 

Using Perfect Squares

We know that when we square a surd, the answer is always going to be rational. This is the key we need to rationalise fractions such as $\frac{8}{3\sqrt{7}}$837.

What do you think will happen if we multiply the denominator here by $\sqrt{7}$7?

Well, the square root sign will disappear, but we also need to make sure the fraction still has the same value, so let's try multiplying the fraction by $\frac{\sqrt{7}}{\sqrt{7}}=1$77=1, which will not change the value of the fraction at all.

$\frac{8}{3\sqrt{7}}\times\frac{\sqrt{7}}{\sqrt{7}}=\frac{8\times\sqrt{7}}{3\times\sqrt{7}\times\sqrt{7}}$837×77=8×73×7×7

which simplifies to

$\frac{8\sqrt{7}}{3\times\left(\sqrt{7}\right)^2}=\frac{8\sqrt{7}}{3\times7}$873×(7)2=873×7

This answer can again be finally simplified to: $\frac{8\sqrt{7}}{21}$8721

The fraction now looks completely different! But try putting it in your calculator, do you get the same value as $\frac{8}{3\sqrt{7}}$837?

So now we know one way of rationalising the denominator is to multiply top and bottom by the surd in the denominator.

 

Rationalising the denominator using perfect squares

To rationalise the denominator of the form $\frac{a}{b\sqrt{n}}$abn we want to multiply it by the fraction $\frac{\sqrt{n}}{\sqrt{n}}$nn:

$\frac{a}{b\sqrt{n}}\times\frac{\sqrt{n}}{\sqrt{n}}=\frac{a\sqrt{n}}{bn}$abn×nn=anbn

 

Using differences of two squares

The above technique is great for when there is only one term in the denominator, but it doesn't work when we have a binomial expression in the denominator.

A binomial is an expression of the form $A+B$A+B, containing two terms. Changing the sign of the second term gives us the binomial $A-B$AB, which we call a conjugate for the original binomial $A+B$A+B.

If we then try to find a conjugate for the binomial $A-B$AB by changing the sign of the second term, we obtain the original binomial $A+B$A+B. That is, any binomial is a conjugate of its own conjugate. We often refer to two such binomials as a conjugate pair.

Notice that the product of a conjugate pair has a familiar form $\left(A+B\right)\left(A-B\right)$(A+B)(AB) which is the factorised form of the difference of two squares $A^2-B^2$A2B2. This means the expression $A^2-B^2$A2B2 will always be rational even if the terms $A$A or $B$B are surds.

 

Conjugates of binomials with surds

Consider a binomial such as $1+\sqrt{2}$1+2. We can find a conjugate for this expression in the same way - by switching the sign of the second term. Doing so, we find that $1-\sqrt{2}$12 is a conjugate for $1+\sqrt{2}$1+2.

The process is the same even if the expression is more complicated, such as $\sqrt{x}-4\sqrt{3}$x43. A conjugate for this expression would be $\sqrt{x}+4\sqrt{3}$x+43.

 

Congugates

For any binomial expression $A+B$A+B, we can find a conjugate $A-B$AB by changing the sign of the second term.

A binomial and its conjugate are sometimes called a conjugate pair.

A side note

We can rewrite the binomial $A+B$A+B in the equivalent form $B+A$B+A by changing the order of the terms. By doing so we can see that $B-A$BA is also a conjugate for this expression, as well as $A-B$AB.

That is, a binomial has two possible conjugates (since there are two orders in which the binomial can be written).

 

Worked example

Rationalise the denominator of $\frac{5}{\sqrt{6}-1}$561.

Think: If we use the perfect squares technique and multiply the bottom by $\sqrt{6}$6, it'll become $6-\sqrt{6}$66. So whilst it got rid of the original $\sqrt{6}$6, it has turned the $1$1 into a $\sqrt{6}$6

We want to multiply the numerator and denominator by the conjugate of the denominator $\sqrt{6}-1$61. We want to change the sign of this expression to find the conjugate, that is, $\sqrt{6}+1$6+1, so let's multiply the expression by $\frac{\sqrt{6}+1}{\sqrt{6}+1}$6+16+1.

 

Do:

$\frac{5}{\sqrt{6}-1}$561 $=$= $\frac{5}{\sqrt{6}-1}\times\frac{\sqrt{6}+1}{\sqrt{6}+1}$561×6+16+1

Multiplying by the conjugate of the denominator

  $=$= $\frac{5\times\left(\sqrt{6}+1\right)}{\left(\sqrt{6}-1\right)\times\left(\sqrt{6}+1\right)}$5×(6+1)(61)×(6+1)

Combinging the fractions

  $=$= $\frac{5\sqrt{6}+5}{\left(\sqrt{6}\right)^2+\sqrt{6}-\sqrt{6}-1}$56+5(6)2+661

Expanding the brackets

  $=$= $\frac{5\sqrt{6}+5}{6-1}$56+561

Simplifying the expression in the denominator

  $=$= $\frac{5\sqrt{6}+5}{5}$56+55

Evaluating the difference in the denominator

  $=$= $\sqrt{6}+1$6+1

Cancelling common factors

 

Reflect: By rationalising the denominator we have in fact gotten rid of the denominator entirely, and we are left with the much simpler expression of $\sqrt{6}=1$6=1.

 

Rationalising the denominator using conjugates

To rationalise the denominator of the form $\frac{a}{b\sqrt{m}+c\sqrt{n}}$abm+cn we want to multiply it by the fraction $\frac{b\sqrt{m}-c\sqrt{n}}{b\sqrt{m}-c\sqrt{n}}$bmcnbmcn:

$\frac{a}{b\sqrt{m}+c\sqrt{n}}\times\frac{b\sqrt{m}-c\sqrt{n}}{b\sqrt{m}-c\sqrt{n}}=\frac{a\left(b\sqrt{m}-c\sqrt{n}\right)}{b^2m-c^2n}$abm+cn×bmcnbmcn=a(bmcn)b2mc2n

 

Practice questions

Question 1

Express the following fraction in simplest surd form with a rational denominator::

$\frac{2}{\sqrt{6}}$26

Question 2

Simplify $\frac{\sqrt{39}+\sqrt{6}}{\sqrt{3}}$39+63.

Question 3

Express the following fraction in simplest surd form with a rational denominator:

$\frac{3}{5\sqrt{2}-4}$3524

Outcomes

VCMNA355 (10a)

Define rational and irrational numbers and perform operations with surds and fractional indices.

What is Mathspace

About Mathspace